Coupled-inductor core for unbalanced phase currents

ABSTRACT

An embodiment of a coupled-inductor core includes first and second members and first and second forms extending between the first and second members. The first form has a parameter (e.g., length) of a first value, and is operable to conduct a first magnetic flux having a first density that depends on the first value of the parameter. The second form is spaced apart from the first form, has the parameter (e.g., length) of a second value different from the first value, and is operable to conduct a second magnetic flux having a second density that depends on the second value of the parameter. Because two or more of the forms of such a core may have different values for the same parameter, the core may be suitable for use in a multiphase power supply where the currents through the phases are unbalanced.

CLAIM OF PRIORITY

The present application is a divisional of U.S. patent application Ser. No. 12/136,018, filed on Jun. 9, 2008, which claims priority to U.S. Provisional Application Ser. No. 60/933,949 filed on Jun. 8, 2007, these applications are incorporated herein by reference in their entireties.

CROSS-RELATED APPLICATIONS

This application is related to U.S. application Ser. No. 12/136,014, filed on Jun. 9, 2008, now U.S. Pat. No. 8,570,009, issued Oct. 29, 2013 and Ser. No. 12/136,023, filed on Jun. 9, 2008, now U.S. Pat. No. 8,179,116, issued May 15, 2012 which have a common owner and which are incorporated herein by reference.

SUMMARY

This Summary is provided to introduce, in a simplified form, a selection of concepts that are further described below in the Detailed Description. This Summary is not intended to identify key features or essential features of the claimed subject matter, nor is it intended to be used to limit the scope of the claimed subject matter.

An embodiment of a coupled-inductor core includes first and second members and first and second forms extending between members. The first form has a parameter (e.g., length) of a first value, and is operable to conduct a first magnetic flux having a first density that depends on the first value of the parameter. The second form is spaced apart from the first form, has the parameter (e.g., length) of a second value different from the first value, and is operable to conduct a second magnetic flux having a second density that depends on the second value of the parameter.

Because two or more of the forms of such a core may have different values for the same parameter, the core may be suitable for use in a multiphase power supply where the currents through the phases are unequal, i.e., are unbalanced, but the windings disposed about the forms have the same number of turns. As compared to a coupled-inductor assembly—“coupled-inductor assembly” is an assembly that includes the core and the windings—having windings with different numbers of turns, a coupled-inductor assembly having windings with the same number of turns may have at least one winding with less resistance, and thus may be more energy efficient and may generate less heat.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a schematic diagram of an embodiment of a multiphase buck converter that includes a coupled-inductor assembly.

FIG. 2 is a diagram of switching signals that drive, and balanced currents that flow through, the phase paths of an embodiment of the buck converter of FIG. 1.

FIG. 3 is schematic diagram of an embodiment of a phase-current unbalancing circuit that may compose part of the power-supply controller of FIG. 1.

FIG. 4 is a perspective view of an embodiment of the coupled-inductor assembly of FIG. 1 that has at least two forms with different values for the same form parameter.

FIG. 5 is a magnetic-circuit model of an embodiment of the coupled-inductor assembly of FIG. 4.

FIG. 6 is a simplified magnetic-circuit model of an embodiment of the coupled-inductor assembly of FIG. 4 having two wound forms, a leakage form, and two windings respectively disposed about the two winding forms.

FIG. 7 is a block diagram of an embodiment of a computer system having a multiphase power supply that includes an embodiment of the coupled-inductor assembly of FIG. 4.

FIG. 8 is a perspective view of an embodiment of a portion of the coupled-inductor assembly of FIG. 4.

DETAILED DESCRIPTION

FIG. 1 is a schematic diagram of an embodiment of a multiphase buck converter 10, which includes phase paths (alternatively “phases”) 12 ₁-12 _(n) and a coupled-inductor assembly 14 having magnetically coupled windings 16 ₁-16 _(n), one winding per phase. As discussed below in conjunction with FIGS. 3-6, the assembly 14 may be designed to carry unbalanced phase currents even when the windings 16 carrying these currents have the same number N of turns. This may reduce the resistance of at least one of these windings, and thus may allow the converter 10 to have an increased power efficiency and to generate less heat as compared to a buck converter that incorporates a conventional coupled-inductor assembly having windings with different numbers of turns.

In addition to the coupled-inductor assembly 14, the converter 10 includes a power-supply controller 18, high-side drive transistors 20 ₁-20 _(n), low-side drive transistors 22 ₁-22 _(n), a filter capacitor 24, and an optional filter inductor 26. A winding 16 and the high-side and low-side transistors 20 and 22 coupled to the winding compose a respective phase 12. For example, the winding 16 ₁ and transistors 20 ₁ and 22 ₁ compose the phase 12 ₁.

The controller 18 may be any type of controller suitable for use in a buck converter, is supplied by voltages VDD_(controller) and VSS_(controller), and receives the regulated output voltage V_(out), a reference voltage V_(ref), and feedback signals I_(FB1)-I_(FBn), which are respectively proportional to the phase currents i₁-i_(n) that flow through the respective phase windings 16 ₁-16 _(n). For example, each of these feedback signals I_(FB1)-I_(FBn) may be a respective voltage that has substantially the same signal phase as the corresponding phase current i and that has an amplitude proportional to the amplitude of the corresponding phase current, and may be generated by a series coupled resistor and capacitor (not shown in FIG. 1) coupled in electrical parallel with the corresponding winding 16.

The high-side transistors 20 ₁-20 _(n), which are each switched “on” and “off” by the controller 18, are power NMOS transistors that are respectively coupled between input voltages VIN₁-VIN_(n) and the windings 16 ₁-16 _(n). Alternatively, the transistors 20 ₁-20 _(n) may be other than power NMOS transistors, and may be coupled to a common input voltage. Moreover, the transistors 20 ₁-20 _(n) may be integrated on the same die as the controller 18, may be integrated on a same die that is separate from the die on which the controller is integrated, or may be discrete components.

Similarly, the low-side transistors 22 ₁-22 _(n), which are each switched on and off by the controller 18, are power NMOS transistors that are respectively coupled between low-side voltages VL₁-VL_(n) and the windings 16 ₁-16 _(n). Alternatively, the transistors 22 ₁-22 _(n) may be other than power NMOS transistors, and may be coupled to a common low-side voltage such as ground. Moreover, the transistors 22 ₁-22 _(n) may be integrated on the same die as the controller 18, may be integrated on a same die that is separate from the die on which the controller is integrated, may be integrated on a same die as the high-side transistors 20 ₁-20 _(n), may be integrated on respective dies with the corresponding high-side transistors 20 ₁-20 _(n) (e.g., transistors 20 ₁ and 22 ₁ on a first die, transistors 20 ₂ and 22 ₂ on a second die, and so on), or may be discrete components.

The filter capacitor 24 is coupled between V_(out) and a voltage VSS_(cap), and works in concert with the windings 16 ₁-16 ₁, and the filter inductor 26 (if present) to maintain the amplitude of the steady-state ripple voltage component of the regulated output voltage V_(out) within a desired range that may be on the order of hundreds of microvolts to tens of millivolts. Although only one filter capacitor 24 is shown, the converter 10 may include multiple filter capacitors coupled in electrical parallel. Furthermore, VSS_(cap) may be equal to VSS_(controller) and to VL₁-VL_(n); for example, all of these voltages may equal ground.

As further discussed below, the filter inductor 26 may be omitted if the leakage inductances L_(lk1)-L_(lkn) of the windings 16 ₁-16 _(n) are sufficient to perform the desired inductive filtering function. In some applications, the filter inductor 26 may be omitted to reduce the size and component count of the converter 10.

Each of the windings 16 ₁-16 _(n) of the coupled-inductor assembly 14 may be modeled as a self inductance L and a resistance DCR. For purposes of discussion, only the model components of the winding 16 ₁ are discussed, it being understood that the model components of the other windings 16 ₂-16 _(n) are similar, except for possibly their values.

The self inductance L₁ of the winding 16 ₁ may be modeled as two zero-resistance inductances: a magnetic-coupling inductance LC₁, and a leakage inductance L_(lk1). When a phase current i₁ flows through the winding 16 ₁, the current generates a magnetic flux. The value of the coupling inductance LC₁ is proportional to the amount of this flux that is coupled to other windings 16 ₂-16 _(n), and the value of the leakage inductance L_(lk1) is proportional to the amount of the remaining flux, which is not coupled to the other windings 16 ₂-16 _(n). In one embodiment, LC₁=LC₂= . . . =LC_(n), and L_(lk1)=L_(lk2)= . . . =L_(lkn), although inequality among the coupling inductances LC, the leakage inductances L_(lk), or both LC and L_(lk), is contemplated. Furthermore, in an embodiment, the respective magnetic-coupling coefficients between pairs of coupling inductances LC are equal (e.g., a current through LC₁ magnetically induces respective equal currents in LC₂, . . . LC_(n)), although unequal coupling coefficients are contemplated.

The resistance DCR₁ is the resistance of the winding 16 ₁ when a constant voltage V is applied across the winding and causes a constant current I to flow through the winding. That is, DCR₁=V/I.

FIG. 2 is a timing diagram of the drive signals PWM1 and PWM2 and the phase currents i₁ and i₂ during steady-state operation of an embodiment of the converter 10 of FIG. 1 having two magnetically coupled phases 16 ₁ and 16 ₂, where the phase currents i₁ and i₂ are balanced. The signals, however, may not be drawn to scale.

Because the phase currents i₁ and i₂ are balanced, the maximum A_(1max) of i₁ substantially equals the maximum A_(2max) of i₂, the minimum A_(1min) of i₁ substantially equals the minimum A_(2min) of i₂, the peak-to-peak amplitude A_(1pp) substantially equals the peak-to-peak amplitude A_(2pp) of i₂, and the average A_(1avg) of i₁ substantially equals the average A_(2avg) of i₂.

Referring to FIGS. 1-2, an embodiment of the operation of the buck converter 10 is discussed where the converter has only two phases 12 ₁ and 12 ₂. In this embodiment, the two phases 12 ₁ and 12 ₂ are switched 180° apart at a switching frequency F_(sw).

While PWM1 is high (logic 1), the high-side transistor 20 ₁ is on and the low-side transistor 22 ₁ is off, and an increasing phase current i₁ flows from VIN₁, through the transistor 20 ₁, winding 16 ₁, and filter inductor 26 (if present), and to the capacitor 24 and to a load 28 that is supplied by V_(out). This increasing current i₁ generates a magnetic flux that induces a corresponding increase in the current i₂ flowing through the other coupled phase 12 ₂.

In contrast, while PWM1 is low (logic 0), the high-side transistor 20 ₁ is off and the low-side transistor 22 ₁ is on, and the current i₁, which is now decreasing, flows from VL₁, through the transistor 22 ₁, winding 16 ₁ and filter inductor 26 (if present), and to the capacitor 24 and to the load 28. The current i₂ also decreases. The currents i₁ and i₂ continue to decrease until PWM2 goes high as discussed below.

Similarly, while PWM2 is high, the high-side transistor 20 ₂ is on and the low-side transistor 22 ₂ is off, and an increasing current i₂ flows from VIN₂, through the transistor 20 ₂, winding 16 ₂, and filter inductor 26 (if present), and to the capacitor 24 and to the load 28. This increasing current i₂ generates a magnetic flux that induces a corresponding increase in the current i₁ flowing through the phase 12 ₁.

In contrast, while PWM2 is low, the high-side transistor 20 ₂ is off and the low-side transistor 22 ₂ is on, and the current i₂, which is now decreasing, flows from VL₂, through the transistor 22 ₂, winding 16 ₂ and filter inductor 26 (if present), and to the capacitor 24 and to the load 28. The current i₁ also decreases. The currents i₁ and i₂ continue to decrease until PWM1 goes high again as discussed above.

The controller 18 compares V_(out) to V_(ref), and controls the duty cycles of PWM1 and PWM2 to maintain a predetermined constant relationship between V_(out) and V_(ref) in the steady state, e.g., V_(out)=2V_(ref) for example, V_(out) may be the average or minimum of the output ripple component. For example, as current drawn by the load 28 increases, the controller 18 may increase the on times or otherwise alter the duty cycles of PWM1 and PWM2 to accommodate the increased load current; conversely, as the load current decreases, the controller may decrease the on times or otherwise alter the duty cycles of PWM1 and PWM2 to accommodate the decreased load current. The controller 18 may use a pulse-width-modulation (PWM) technique, a constant-on-time technique, or another technique to control the on and off times of the high-side and low-side transistors 20 ₁-20 ₂ and 22 ₁-22 ₂.

The controller 18 may also monitor the feedback signals I_(FB1) and I_(FB2), and adjusts the on times or otherwise adjusts the duty cycles of PWM1 and PWM2 as needed to maintain the balance between the currents i₁ and i₂. That is, the controller 18 adjusts PWM1 and PWM2 so that A_(1max), A_(1min), A_(1pp), and A_(1avg) stay substantially equal to A_(2max), A_(2min), A_(2pp), and A_(2avg) respectively. An embodiment of current-balancing circuitry that the controller 18 may include is disclosed in U.S. Pat. No. 6,278,263, which is incorporated by reference.

Because the currents i₁ and i₂ are balanced, the coupled-inductor assembly 14 may be symmetrical such that there is little or no chance of saturating the core (not shown in FIGS. 1-2) of the assembly. That is, the core rung about which the winding 16 ₁ is wound has the same parameters (e.g., dimensions, reluctance) as the rung about which the winding 16 ₂ is wound, and the number N₁ of turns of the winding 16 ₁ is the same as the number N₂ of turns of the winding 16 ₂. For an ideal, symmetrical assembly 14 carrying balanced currents i₁ and i₂ and having L_(lk1)=L_(lk2)=0, the net flux through the core is zero, and thus there is no danger of saturating the core. And even if the assembly 14 is not ideal such that L_(lk1) and L_(lk2) are non zero, the peak magnitudes of the fluxes flowing through the core may be small enough such that there is little chance of saturating the core.

But still referring to FIGS. 1-2 and as discussed below in conjunction with FIGS. 3-6, one may design the converter 10 so that two or more of the windings 16 ₁-16 _(n) are able to carry a respective two or more unbalanced currents i₁-i_(n). Unbalanced currents i flowing in the phases 12 may be suitable, for example, where the controller 18 may deactivate one or more of the phases 12 under less-than-full load conditions to conserve power. For example, consider a two-phase embodiment of the converter 10 that is designed to provide a steady-state average light load current of 15 Amperes (A), a steady-state average normal-load current of 30 A, and a steady-state average heavy-load current of 45 A. During light-load conditions, the controller 18 activates only the phase 12 ₁, and causes this phase to provide a 15 A steady-state average current i₁. Similarly, during normal-load conditions, the controller 18 activates only the phase 12 ₂, and causes this phase to provide a 30 A steady-state average current i₂. And during heavy-load conditions, the controller 18 activates both phases 12 ₁ and 12 ₂, and causes these phases to together provide a 45 A steady-state average current (i_(avg)=15 A and i_(2avg)=30 A). Although this example discusses only the unbalanced currents i₁ and i₂ having unequal average amplitudes A_(avg1) and A_(avg2), two phase currents, e.g., the currents i₁ and i₂, may be unbalanced if one or more of A_(max), A_(min), A_(pp), and A_(avg) of the first current do not substantially equal a respective one or more of A_(max), A_(min), A_(pp), and A_(avg) of the second current. An embodiment of a circuit for unbalancing phase currents is discussed below in conjunction with FIG. 3.

If the converter 10 is designed for two or more windings 16 ₁-16 _(n) to carry unbalanced currents, then the core of the coupled-inductor assembly 14 may be designed so that no portion of the assembly's core saturates during operation of the converter.

One technique for preventing saturation of the assembly 14 core is to make the Ni products equal in each of the phases carrying unbalanced currents i; therefore, because the currents i in these phases are unequal, the number of turns N of the respective windings 16 are also unequal.

But this technique may result in one or more of the corresponding windings 16 being longer than, and thus having a greater DCR than, respective windings in a core assembly designed to carry balanced currents i.

Furthermore, because N must be an integer per Maxwell's equations, this technique constrains the unbalanced currents i to be integer multiples of one another.

Still referring to FIGS. 1-2, however, another technique for preventing saturation of the core may reduce the DCR of at least one winding 16 as compared to the previously described technique, and may allow one unbalanced current to be a non-integer multiple of another unbalanced current. In an embodiment of this other technique, portions of the core, such as the rungs about which the windings are wound, have different values of a same parameter, and each of the respective windings 16 carrying the unbalanced currents has a same number N of turns, such that one winding need not be longer than another winding as in the previously described technique. But this technique does contemplate that the respective windings carrying the unbalanced currents may have different numbers N of turns.

Reducing the DCR of one or more of the windings 16 ₁-16 _(n) may reduce the amount of power (I_(rms) ²·DCR) that the windings (and thus the coupled-inductor assembly 14) consume, and thus may reduce the amount of heat that the windings (and thus the coupled-inductor assembly) generate.

Consequently, a coupled-inductor assembly 14 allowing unbalanced phase currents and having one or more windings with reduced DCRs may allow the converter 10 to be more power efficient and to generate less heat than a converter that has unbalanced phase currents and that includes a coupled-inductor assembly with windings having different numbers of turns.

Furthermore, allowing one unbalanced current to be a non-integer multiple of another unbalanced current may increase the number of applications in which one may use a coupled-inductor power supply such as the converter 10.

Referring again to FIG. 1, alternate embodiments of the buck converter 10 are contemplated. For example, the converter 10 may be modified to generate V_(out) having a negative value. Furthermore, the converter 10 may include one or more magnetically uncoupled phases as described in previously incorporated U.S. patent application Ser. Nos. 12/136,014 and 12/136,023.

Further descriptions of coupled-inductor power supplies appear in the following references, which are incorporated by reference: Wong et al., Investigating Coupling Inductors In The Interleaved QSW VRM, IEEE 2000, and Park et al., Modeling And Analysis Of Multi-Interphase Transformers For Connecting Power Converters In Parallel, IEEE 1997.

FIG. 3 is a diagram of an embodiment of a current-unbalancing circuit 30, which the controller 18 of FIG. 1 may include to unbalance two or more of the phase currents i₁-i_(n) of the buck converter 10 of FIG. 1.

The circuit 30 includes an error amplifier 32, a phase-current summer 34, phase-current scalers 36 ₁-36 _(n), current-unbalancing summers 38 ₁-38 _(n), gain blocks 40 ₁-40 _(n), PWM summers 42 ₁-42 _(n), and PWM signal generators 44 ₁-44 _(n).

The error amplifier 32 generates a control voltage V_(EA), which is proportional to the difference between the regulated output voltage V_(nut) (or a scaled version of V_(out)) and the reference voltage V_(ref). V_(EA) may be the result of the amplifier 32 low-pass filtering the difference between V_(out) and V_(ref).

The phase-current summer 34 adds the feedback signals I_(FB1)-I_(FBn) to generate a signal I_(LOADcalculated), which is proportional to the actual load current I_(LOAD) through the load 28 (FIG. 1).

Each of the phase-current scalers 36 ₁-36 _(n) scales I_(LOADcalculated) by a respective one of the values n/a₁-n/a_(n) according to the respective phase current to be carried by the respective winding 16, where n is the number of coupled phases 12 ₁-12 _(n) and where a₁-a_(n) are respective balancing coefficients.

Each of the current-unbalancing summers 38 ₁-38 _(n) subtracts from a respective signal I_(FB1)-I_(FBn) a scaled signal from a respective one of the scalers 36 ₁-36 _(n).

Each of the gain blocks 40 ₁-40 _(n) multiplies a sum from a respective one of the summers 38 ₁-38 _(n) by a respective gain value G₁-G_(n) to generate a respective signal ΔI₁-ΔI_(n). The values G₁-G_(n) may be selected to give specified frequency and error-correction responses to the respective feedback loops formed in part by the summer 34, the scalers 36 ₁-36 _(n), the summers 38 ₁-38 _(n), and the gain blocks 40 ₁-40 _(n). Alternatively, the gain blocks 40 ₁-40 _(n) may be omitted, which is equivalent to G₁=G₂= . . . =G_(n)=1.

Each of the summers 42 ₁-42 _(n) subtracts a respective one of the signals ΔI₁-ΔI_(n) from V_(EA).

Each of the PWM signal generators 44 ₁-44 _(n) includes a comparator that receives on a non-inverting input node the resulting sum from a respective one of the summers 42 ₁-42 _(n), that receives on an inverting input node a respective conventional ramp signal, and that generates on an output node a respective one of the signals PWM1-PWMn. The phases of the ramp signals may be offset in a conventional manner such that the phases of PWM1-PWMn are offset, for example, by 360°/n as discussed above in conjunction with FIG. 2.

Referring to FIGS. 1 and 3, the operation of the unbalancing circuit 30 is described according to an embodiment where the converter 10 includes two phases 12 ₁-12 ₂ (n=2), and is designed to generate i₁=3·i₂. In this embodiment, i₁ represents the average current flowing through the phase 12 ₁ and i₂ represents the average current flowing through the phase 12 ₂. But in other embodiments, i₁ and i₂ may represent, e.g., the peak, minimum, or rms currents respectively flowing through the phases 12 ₁ and 12 ₂.

The “on” portions of PWM1 and PWM2 (e.g., the portions of PWM1 and PWM2 at logic 1 in FIG. 2) are proportional to V_(EA). For example, if V_(out) decreases to less than V_(ref), then V_(EA) tends to increase, and this increase in V_(EA) tends to increase the lengths of the on portions of PWM1 and PWM2. The increases in the lengths of the on portions of PWM1 and PWM2 increase the on times of the high-side transistors 20 ₁-20 ₂, and this tends to cause V_(out) to increase back toward V_(ref). Conversely if V_(out) increases to more than V_(ref), then V_(EA) tends to decrease, and this reduction in V_(EA) tends to decrease the lengths of the on portions of PWM1 and PWM2. The decreases in the lengths of the on portions of PWM1 and PWM2 decrease the on times of the high-side transistors 20 ₁-20 ₂, and this tends to cause V_(out) to decrease back toward V_(ref).

As described below, the signals ΔI₁-ΔI₂ effectively and respectively adjust the signal V_(EA) for the phases 12 ₁-12 ₂ via the PWM summers 42 ₁-42 _(n) to set i₁=3·i₂.

When

i ₁=3·i ₂  (1)

then

I _(FB1)=3·I _(FB2)  (2)

n/a ₁ ·I _(LOADcalculated) =I _(FB1)  (3)

and

n/a ₂ ·I _(LOADcalculated) =I _(FB2)  (4)

Also, because n/a₁·I_(LOADcalculated)+n/a₂·I_(LOADcalculated)=I_(LOADcalculated), then

n/a ₁ +n/a ₂=1.  (5)

Therefore, from equations (2)-(4), one may derive the following:

n/a ₁ ·I _(LOADcalculated)=3·I _(FB2)=3·n/a ₂ ·I _(LOADcalculated)  (6)

and

a ₂=3a ₁.  (7)

And, from equations (5) and (7) and that n=2 in this embodiment, one may derive the following:

a ₁ =a ₂ ·n/(a ₂ −n)  (8)

a ₁=8/3  (9)

and

a ₂=8.  (10)

Therefore, when I_(FB1)=3·I_(FB2), I_(FB1)=2/(8/3)=¾·I_(LOADcalculated), I_(FB2)=¼·I_(LOADcalculated), the output of the unbalancing summer 38 ₁ equals zero, and the output of the unbalancing summer 38 ₂ equals zero, and the circuit 30 makes no adjustment to the signal V_(EA) from the error amplifier 32.

But suppose that I_(FB1)<3·I_(FB2), and thus both I_(FB1), and i₁ are too low.

Therefore, the difference output of the unbalancing summer 38 ₁ is negative and the value ΔI₁ is negative, but the inverting input of the PWM summer 42 ₁ effectively makes ΔI₁ positive, and thus makes the output of the PWM summer 44 ₁ more positive as compared to the summer output due to V_(EA) alone. This tends to increase the on portion of PWM1, and thus tends to increase both I_(FB1), and i₁.

In contrast, when I_(FB1)<3·I_(FB2), then I_(FB2)>(I_(FB1)/)3, and both I_(FB2) and i₂ are too high.

Therefore, the difference output of the unbalancing summer 38 ₂ is positive and the value ΔI₂ is positive, but the inverting input of the PWM summer 42 ₂ effectively makes ΔI₂ negative, and thus makes the output of the PWM summer 44 ₂ more negative as compared to the output due to V_(EA) alone. This tends to decrease the on portion of PWM2, and thus tends to decrease both I_(FB2) and i₂.

Consequently, when I_(FB1) is too low and I_(FB2) is too high, the current unbalancing circuit 30 acts to increase I_(FB1) and reduce I_(FB2) to the values set by the scalers 36 ₁-36 ₂, and thus acts to increase i₁ and reduce i₂ toward i₁=3·i₂. But since the current-unbalancing circuit 30 maintains I_(LOADcalculated) (and thus maintains the actual load current I_(LOAD)) substantially constant, the unbalancing circuit ideally does not cause V_(out) to drift out of regulation by causing V_(out) to increase or decrease relative to V_(ref).

In a similar manner, the unbalancing circuit 30 acts to decrease I_(FB1) and i₁ and to increase I_(FB2) and i₂ to the respective values set by the scalers 36 ₁-36 ₂ when I_(FB1) and i₁ are too high and I_(FB2) and i₂ are too low.

And when I_(FB1) and I_(FB2) equal the values respectively set by the scalers 36 ₁ and 36 ₂ (and thus i₁=3·i₂), the difference signals ΔI₁=ΔI₂=0, and the signal V_(EA) sets the on portions of PWM1 and PWM2 as if the circuitry generating ΔI₁ and ΔI₂ were not present.

Still referring to FIGS. 1 and 3, the current-unbalancing circuit 30 may operate in a similar manner when the converter 10 includes more than two phases 12, and when the currents i through more than two of these phases are unbalanced.

Referring again to FIG. 3, alternate embodiments of the current-unbalancing circuit 30 are contemplated.

FIG. 4 is a perspective view of an embodiment of the coupled-inductor core assembly 14 of FIG. 1, where the core assembly may be designed for use when two or more of the phases 12 ₁-12 _(n) of the buck converter 10 of FIG. 1 carry unbalanced currents.

In addition to the windings 16 ₁-16 _(n), the core assembly 14 includes a core 50 having winding forms 52 ₁-52 _(n) and an optional leakage form 53, and members 54 and 56, which interconnect the forms. That is, using a ladder analogy, the forms 52 ₁-52 _(n) are the rungs of the ladder, and the members 54 and 56 are the rails to which the rungs are connected. Spaces 58 ₁-58 _(n) are located between the forms 52 ₁-52 _(n) and 54.

Each winding 16 ₁-16 _(n) is formed from a respective conductor 60 ₁-60 _(n), which has a respective width W₁-W_(n), and each conductor 60 ₁-60 _(n) is wound, in a Faraday's law sense, N₁-N_(n) turns about a respective form 52 ₁-52 extends beneath and adjacent to the remaining forms. For example, the winding 16 ₁ is formed from a conductor 60 ₁ that is wound N₁=1 turn about the form 52 ₁ and extends beneath and adjacent to the remaining forms 52 ₂-52 _(n). Although the conductor 60 ₁ is adjacent to only three sides of the form 52 ₁, the current i₁ through the conductor 60 ₁ traverses a closed loop through which the form 52 ₁ passes. The portions of this closed loop other than the conductor 60 ₁ may be formed, e.g., by a conductive trace on a circuit board on which the core assembly 14 is disposed. Therefore, N₁ is the integer number of closed loops through which the form 52 ₁ extends. Similarly, the winding 16 ₂ is formed from a conductor 60 ₂ that is wound N₂=1 turn about the form 52 ₂ and extends beneath and adjacent to the remaining forms 52 ₁ and 52 ₃-52 _(n), and so on. The conductors 60 ₁-60 _(n) may be made from any suitable conductive material such as copper or another metal, and may, but need not be, electrically insulated from the forms 52 ₁-52 _(n).

Each form 52 ₁-52 _(n) and 53 has a respective length I₁-I_(n) and I_(lk), a respective permeability μ1-μ_(n) and μ_(lk), and a respective cross-sectional area A₁-A_(n) and A_(lk). Although some of these like parameters (e.g., the lengths I₁-I_(n) and I_(lk)) are shown as being equal in FIG. 4, some or all of these like parameters may have different values. Furthermore, some or all of the permeabilities μ₁-μ_(n) and μ_(lk) may be made to be different by forming the respective forms 52 from different materials. Moreover, an optional gap 62 (e.g., an air gap) may be disposed in the leakage form 53 to adjust the reluctance thereof. Alternatively, the optional gap 62 may be distributed throughout the material from which the leakage form 53 is made.

Referring to FIGS. 1 and 4, the operation of the coupled-inductor assembly 14 is generally described when a current i₁ flows through the conductor 60 ₁ in the direction shown, it being understood that the operation is similar when a current flows through the other conductors 60 ₂-60 _(n). For purposes of example, it is assumed that the coupled-inductor assembly 14 is mounted to a printed circuit board such that the forms 52 ₂-52 _(n) do not pass inside the loop(s) composed in part by the conductor 60 ₁.

As the current i₁ flows through the conductive loop composed in part by the conductor 60 ₁, it generates a total magnetic flux φ_(T). In a first-order approximation, a first portion φ₁ of the total flux φ_(T) flows through the form 52 ₁, and a second portion φ_(out) of the total flux φ_(T) flows outside of the form 52 ₁ such that φ_(T) is given by the following equation:

φ_(T)=φ₁+φ_(out)  (11)

The first flux portion φ₁ flows through the remaining forms 52 ₂-52 _(n) and 53 such that the sum of the fluxes φ₂-φ_(n) and φ_(lk) flowing through each of the remaining forms equals φ₁. Because the portion of φ₁ that equals φ₂+φ₃+ . . . +φ_(n) flows through the forms 52 ₂-52 _(n) about which the conductors 60 ₂-60 _(n) are wound, this portion of φ₁ is called the coupling flux. The portion φ₂-φ_(n) of the coupling flux φ₁ induce respective currents to flow in the conductors 60 ₂-60 _(n). As discussed further below in conjunction with FIGS. 5-6, the specific values of the fluxes φ₁-φ_(n) depend on the reluctances R₁-R_(n) and R_(lk) of the forms 52 ₁-52 _(n) and 53.

And the remaining flux equal to the sum φ_(out)+φ_(lk) is called the total leakage flux, because it does not induce any currents to flow in the conductors 60 ₂-60 _(n). Where φ_(out)<<φ_(lk), then the total leakage flux may be approximated as φ_(lk). As discussed further below in conjunction with FIGS. 5-6, the specific value of the flux φ_(lk) depends on the reluctances R₁-R_(n) and R_(lk) of the forms 52 ₁-52 _(n) and 53.

Referring again to FIG. 4, each portion of the core 50 has a respective maximum flux density B above which that portion of the core will magnetically saturate.

For known reasons that are omitted for brevity, it may be desirable to run a power supply like the buck converter 10 of FIG. 1 such that no portion of the core 50 saturates when the phases 12 ₁-12 _(n) carry the respective currents i₁-i_(n) that they are designed to carry.

An embodiment of a technique for designing the core 50 and the core-assembly 14 so that no portion of the core saturates under expected operating conditions is discussed below in conjunction with FIGS. 5-6. For reasons discussed above in conjunction with FIG. 1, this embodiment may allow N₁≠N₂≠ . . . ≠N_(n).

FIG. 5 is an embodiment of an equivalent magnetic circuit 70 of the core-assembly 14 of FIG. 4. R₁-R_(n) and R_(lk) are the respective reluctances of the forms 52 ₁-52 _(n) and 53 of FIG. 4, R₁₂-R_((n-1)n) and R_(nk) are the respective reluctances of the portions of the member 54 between adjacent pairs of the forms, and R₂₁-R_(n(n-1)) and R_(kn) are the respective reluctances of the portions of the member 56 between adjacent pairs of the forms. For example, R₁₂ is the reluctance of the portion of the member 54 between the forms 52 ₁ and 52 ₂, R₂₃ is the reluctance of the portion of the member 54 between the forms 52 ₂ and 52 ₃, and so on. Similarly, R₂₁ is the reluctance of the portion of the member 56 between the forms 52 ₂ and 52 ₁, R₃₂ is the reluctance of the portion of the member 56 between the forms 52 ₃ and 52 ₂, and so on. Also, the directions of the fluxes φ₂-φ_(n) have been reversed relative to their directions in FIG. 4 for purposes of mathematical convention that may simplify mathematical calculations that use the circuit model 70. Furthermore, the model 70 may ignore the leakage flux φ_(out) if φ_(out)<<φ_(lk), or may include the contribution of φ_(out) in φ_(lk).

FIG. 6 is an embodiment of an equivalent magnetic circuit 80 of the core-assembly 14 of FIG. 4 for a two-phase embodiment (n=2) of the buck converter 10 of FIG. 1. In this embodiment, it is assumed that the reluctances of the members 54 and 56 are much less (for example, at least ten times less) than each of the reluctances R₁-R₂ and R_(lk); therefore, the reluctances of the members 54 and 56 are approximated as zero in the magnetic circuit 80, thus further simplifying the circuit 80 as compared to the circuit 70 of FIG. 5. Furthermore, the circuit model 80 includes the contribution of φ_(out) (FIG. 4) in φ_(lk) by including in R_(lk) the reluctance of the path traversed by φ_(out) (this also assumes that φ_(out) is the same for i₁ and i₂). Moreover, it is assumed that the maximum flux densities of the leakage form 53 and the members 54 and 56 are large enough so that if the forms 52 ₁ and 52 ₂ do not saturate, then the leakage form and the members do not saturate. For example, this assumption may hold true where the leakage form 53 includes the gap 62 (FIG. 4) and the cross-sectional areas of the forms 54 and 56 are significantly larger than the cross-sectional areas A₁ and A₂ of the forms 52 ₁ and 52 ₂.

φ₁ and φ₂ of the model circuit 80 may be given by the following:

$\begin{matrix} {\phi_{1} = \frac{{{N_{1}\left( {R_{2} + R_{lk}} \right)}i_{1}} - {N_{2}R_{lk}i_{2}}}{{R_{1}R_{2}} + {R_{1}R_{lk}} + {R_{2}R_{lk}}}} & (12) \\ {\phi_{2} = \frac{{{- N_{1}}R_{lk}i_{1}} + {{N_{2}\left( {R_{1} + R_{lk}} \right)}i_{2}}}{{R_{1}R_{2}} + {R_{1}R_{lk}} + {R_{2}R_{lk}}}} & (13) \end{matrix}$

And R₁, R₂, and R_(lk) are given by the following:

$\begin{matrix} {R_{1} = \frac{l_{1}}{\mu_{1}A_{1}}} & (14) \\ {R_{2} = \frac{l_{2}}{\mu_{2}A_{2}}} & (15) \\ {R_{lk} = \frac{l_{lk}}{\mu_{lk}A_{lk}}} & (16) \end{matrix}$

Furthermore, the flux densities B₁ and B₂ through the forms 52 ₁ and 52 ₂ are given by the following:

$\begin{matrix} {B_{1} = \frac{\phi_{1}}{A_{1}}} & (17) \\ {B_{2} = \frac{\phi_{2}}{A_{2}}} & (18) \end{matrix}$

Therefore, to prevent saturation of the forms 52 ₁ and 52 ₂ (and thus to prevent saturation of all portions of the core 50 in this example), one may designs the core 50 according to the following:

B ₁ <B _(1max)  (19)

B ₂ <B _(2max)  (20)

where B₁max is the maximum flux density that the form 52 ₁ is able to conduct without saturating, and B_(2max) is the maximum flux density that the form 52 ₂ is able to conduct without saturating. One may determine B_(1max) and B_(2max) in a conventional manner. For example, one may determine B_(1max) and B_(2max) from μ₁ and μ₂, or retrieve B_(1max) and B_(2max) from a table based on the respective materials from which the forms 52 ₁ and 52 ₂ are made.

Referring to FIGS. 4 and 6, an example procedure for designing a two-phase embodiment of the coupled-inductor assembly 14 using the circuit model 80 is described. In this example,

i ₁ =m·i ₂  (21)

where m may be any positive number.

From equations (12)-(21), one may derive the following expressions of the respective flux densities φ₁ and φ₂ in the forms 52 ₁ and 52 ₂ of the core 14, where φ₁ and φ₂ do not saturate the respective forms 52 ₁ and 52 ₂ or any other part of the core 50 in this example:

$\begin{matrix} {B_{1} = {\frac{\phi_{1}}{A_{1}} = {\frac{i_{1}}{{R_{1}R_{2}} + {R_{1}R_{lk}} + {R_{2}R_{lk}}} \cdot {\quad{\left\lbrack {{N_{1} \cdot \frac{l_{2}}{\mu_{2}A_{1}A_{2}}} + {\left( {N_{1} - \frac{N_{2}}{m}} \right) \cdot \frac{l_{lk}}{\mu_{lk}A_{1}{Ak}}}} \right\rbrack < B_{1{ma}\; x}}}}}} & (22) \\ {B_{2} = {\frac{\phi_{2}}{A_{2}} = {\frac{i_{1}}{{R_{1}R_{2}} + {R_{1}R_{lk}} + {R_{2}R_{lk}}} \cdot {\quad{\left\lbrack {{\frac{N_{2}}{m} \cdot \frac{l_{1}}{\mu_{1}A_{1}A_{2}}} + {\left( {\frac{N_{2}}{m} - N_{1}} \right) \cdot \frac{l_{lk}}{\mu_{lk}A_{2}A_{lk}}}} \right\rbrack < B_{2{ma}\; x}}}}}} & (23) \end{matrix}$

In this design example, the following equalities are also assumed:

μ₁=μ₂=μ  (24)

B _(1max) =B _(2max) =B _(max)  (25)

A ₁ =A ₂ =A _(k) =A  (26)

N ₁ =N ₂ =N  (27)

m=3  (28)

From equations (22)-(26), one may derive the following expressions for B₁ and B₂:

$\begin{matrix} {B_{1} = {\frac{\phi_{1}}{A_{1}} = {{\frac{i_{1}N}{3{A^{2}\left( {{R_{1}R_{2}} + {R_{1}R_{lk}} + {R_{2}R_{lk}}} \right)}} \cdot \left\lbrack {\frac{3l_{2}}{\mu} + \frac{2l_{lk}}{\mu_{lk}}} \right\rbrack} < B_{{ma}\; x}}}} & (27) \\ {B_{2} = {\frac{\phi_{2}}{A_{2\;}} = {{\frac{i_{1}N}{3{A^{2}\left( {{R_{1}R_{2}} + {R_{1}R_{lk}} + {R_{2}R_{lk}}} \right)}} \cdot \left\lbrack {\frac{l_{1}}{\mu} - \frac{2l_{lk}}{\mu_{lk}}} \right\rbrack} < B_{{ma}\; x}}}} & (28) \end{matrix}$

Then, for a specified value for i₁, a designer may select values for A, N, μ, μ_(lk), I₁, I₂, and I_(1k) such that equations (27) and (28) are true. Furthermore, to set B₁ equal to B₂, the designer may select the values for μ, μ_(lk), I₁, I₂, and I_(lk) according to the following equation, which is derived from equations (27) and (28):

$\begin{matrix} {{\frac{3l_{2}}{\mu} + \frac{2l_{lk}}{\mu_{lk}}} = {\frac{l_{1}}{\mu} - \frac{2_{lk}}{\mu_{lk}}}} & (29) \end{matrix}$

Furthermore, in this example, although the coupled-inductor assembly 14 is designed for the windings 16 ₁ and 16 ₂ to carry unbalanced currents i₁ and i₂, the windings have the same number of turns N. Moreover, unlike the number of turns N, the variable m is not constrained to only integer values.

Appendix A includes additional mathematical expressions that are derived from the circuit model 80 of FIG. 6 and that may be useful in designing the coupled-inductor assembly 14 and a power supply such as the buck converter 10 (FIG. 1).

Still referring to FIGS. 4 and 6, other embodiments of the coupled-inductor assembly 14 and the above-described procedure for designing the assembly are contemplated. For example, the design procedure may be extrapolated to design a coupled-inductor assembly having more than two coupled phase 12, and having fewer or more than one leakage form. Furthermore, one may use the circuit model 70 of FIG. 5 to design the assembly 14 by modifying the design equations accordingly. Moreover, one may develop design equations similar to those described above to mathematically demonstrate that other portions of the core 50 (e.g., the members 54 and 56 and the leakage form 53) do not saturate in a specified application. Also, in addition to the coupled forms 52 ₁-52 _(n), the coupled-inductor assembly 14 may include uncoupled forms as discussed in U.S. patent application Ser. Nos. 12/136,014 and 12/136,023, which are incorporated by reference. Furthermore, instead of or in addition to the leakage form 53, the core assembly 14 may include a leakage plate as described in U.S. patent application Ser. No. 11/903,185, which is incorporated by reference.

FIG. 7 is a block diagram of a system 90 (here a computer system), which may incorporate a multiphase power supply 92 (such as the buck converter 10 of FIG. 1) that generates unbalanced phase currents and that includes an embodiment of the coupled-inductor assembly 14 of FIG. 4.

The system 90 includes computer circuitry 94 for performing computer functions, such as executing software to perform desired calculations and tasks. The circuitry 94 typically includes a controller, processor, or one or more other integrated circuits (ICs) 96, and the power supply 92, which provides power to the IC(s) 96. One or more input devices 98, such as a keyboard or a mouse, are coupled to the computer circuitry 94 and allow an operator (not shown) to manually input data thereto. One or more output devices 100 are coupled to the computer circuitry 94 to provide to the operator data generated by the computer circuitry. Examples of such output devices 100 include a printer and a video display unit. One or more data-storage devices 102 are coupled to the computer circuitry 94 to store data on or retrieve data from external storage media (not shown). Examples of the storage devices 102 and the corresponding storage media include drives that accept hard and floppy disks, tape cassettes, compact disk read-only memories (CD-ROMs), and digital-versatile disks (DVDs).

FIG. 8 is a view of a portion of the coupled-inductor assembly 14 of FIG. 4, according to an embodiment.

Referring to FIG. 8, an embodiment of the operation of the coupled-inductor assembly 14 is generally described when unbalanced currents i₁, i₂, and i₃—in this example, unbalanced currents means that at least one of the currents is not equal to at least one of the other currents—respectively flow through the windings 16 ₁, 16 ₂, and 16 ₃ in the respective directions shown, it being understood that the operation is similar when respective currents flow through other groups of one or more windings 16. For example purposes, it is assumed that the coupled-inductor assembly 14 is mounted to a printed circuit board such that the forms 52 ₁-52 _(n) do not pass inside the loop(s) composed in part by windings 16 that are not associated with a respective form. For example, the form 52 ₁ passes inside only the loop composed in part by the winding 16 ₁, the form 52 ₂ passes inside only the loop composed in part by the winding 16 ₂, the form 52 ₃ passes inside only the loop composed in part by the winding 16 ₃, etc. Further for example purposes, it is assumed that there are no forms other than forms 52 ₁-52 ₃.

As the current i₁ flows through the conductive loop composed in part by the winding 16 ₁, it generates a total magnetic flux φ₇₋₁. In a first-order approximation, a first portion φ_(1,1) of the total flux φ_(T1) flows through the form 52 ₁, and a second portion φ_(out1) (not shown in FIG. 8) of the total flux φ_(T1) flows outside of the form 52 ₁ such that φ_(T1) is given by the following equation:

φ_(T1)=φ_(1,1)+φ_(out1)  (30)

The first flux portion φ_(1,1) flows through the remaining forms 52 ₂-52 ₃ such that the sum of the fluxes φ_(1,2) and φ_(1,3) respectively flowing through each of the remaining forms equals φ_(1,1). Because the portion φ_(1,1) that equals φ_(1,2)+φ_(1,3) flows through the forms 52 ₂-52 ₃ about which the windings 16 ₂-16 ₃ are wound, this portion φ_(1,1) is called the coupling flux. The portions φ_(1,2) and φ_(1,3) of the coupling flux φ_(1,1) induce respective currents (these induced currents are not shown in FIG. 8) to flow in the conductors 60 ₂-60 ₃.

Furthermore, as the current i₂ flows through the conductive loop composed in part by the winding 16 ₂, it generates a total magnetic flux φ_(T2). In a first-order approximation, a first portion φ_(2,2) of the total flux φ_(T2) flows through the form 52 ₂, and a second portion φ_(out2) (not shown in FIG. 8) of the total flux φ_(T2) flows outside of the form 52 ₂ such that φ_(T2) is given by the following equation:

φ_(T2)=φ_(2,2)+φ_(out2)  (31)

The first flux portion φ_(2,2) flows through the remaining forms 52 ₁ and 52 ₃ such that the sum of the fluxes φ_(2,1) and φ_(2,3) respectively flowing through each of the remaining forms equals φ_(2,2). Because the portion φ_(2,2) that equals φ_(2,1)+φ_(2,3) flows through the forms 52 ₁ and 52 ₃ about which the windings 16 ₁ and 16 ₃ are wound, this portion φ_(2,2) is called the coupling flux. The portions φ_(2,1) and φ_(2,3) of the coupling flux φ_(2,2) induce respective currents (these induced currents are not shown in FIG. 8) to flow in the conductors 60 ₁ and 60 ₃.

Similarly, as the current i₃ flows through the conductive loop composed in part by the winding 16 ₃, it generates a total magnetic flux φ_(T3). In a first-order approximation, a first portion φ_(3,3) of the total flux φ_(T3) flows through the form 52 ₃, and a second portion φ_(out3) (not shown in FIG. 8) of the total flux φ_(T3) flows outside of the form 52 ₃ such that φ_(T3) is given by the following equation:

φ_(T3)=φ_(3,3)+φ_(out3)  (32)

The first flux portion φ_(3,3) flows through the remaining forms 52 ₁ and 52 ₂ such that the sum of the fluxes φ_(3,1) and φ_(3,2) flowing through each of the remaining forms equals φ_(3,3). Because the portion φ_(3,3) that equals φ_(3,1)+φ_(3,2) flows through the forms 52 ₁ and 52 ₂ about which the windings 16 ₁-16 ₂ are wound, this portion φ_(3,3) is called the coupling flux. The portions φ_(3,1) and φ_(3,2) of the coupling flux φ_(3,3) induce respective currents to flow in the conductors 60 ₁ and 60 ₂.

In the above-described example, the form 52 ₁ has a parameter (e.g., permeability, reluctance, or area as described above in conjunction with FIGS. 4-6) of a first value, and is configured to conduct a first magnetic flux given by the following equation:

φ₅₂ _(_) ₁=φ_(1,1)−φ_(2,1)−φ_(3,1)  (33)

Flux φ₅₂ _(_) ₁ has a density B₅₂ _(_) ₁ that depends on the first value of the parameter as described above in conjunction with FIG. 6.

Furthermore, as evident from equation (33), φ₅₂ _(_) ₁ is a combination (e.g., a sum) of a second magnetic flux φ_(1,1) flowing through the form 52 ₁ in a direction (e.g., bottom-to-top in FIG. 8) and a third magnetic flux −(φ₂+φ_(3,1)) simultaneously flowing through the form 52 ₁ in an opposite direction (e.g., top-to-bottom in FIG. 8).

Furthermore, the form 52 ₂ has the same parameter (e.g., permeability, reluctance, or area as described above in conjunction with FIGS. 4-6) as the form 52 ₁ but of a second value that is different from the first value, and is configured to conduct a fourth magnetic flux given by the following equation:

φ₅₂ _(_) ₂=φ_(2,2)−φ_(1,2)−φ_(3,2)  (34)

Flux φ₅₂ _(_) ₂ has a density B₅₂ _(_) ₂ that depends on the second value of the parameter as described above in conjunction with FIG. 6.

Furthermore, as evident from equation (34), φ₅₂ _(_) ₂ is a combination (e.g., a sum) of a fifth magnetic flux φ_(2,2) flowing through the form 52 ₂ in a direction (e.g., bottom-to-top in FIG. 8) and a sixth magnetic flux −(φ_(1,2)+φ_(3,2)) simultaneously flowing through the form 52 ₂ in an opposite direction (e.g., top-to-bottom in FIG. 8).

And the form 52 ₃ has the same parameter (e.g., permeability, reluctance, or area as described above in conjunction with FIGS. 4-6) as the forms 52 ₁ and 52 ₂ but of a third value that may be the same as, or different than, or both of the first and second values, and is configured to conduct a seventh magnetic flux given by the following equation:

φ₅₂ _(_) ₃=φ_(3,3)−φ_(1,3)−φ_(2,3)  (35)

Flux φ₅₂ _(_) ₃ has a density B₅₂ _(_) ₃ that depends on the third value of the parameter as described above in conjunction with FIG. 6.

Furthermore, as evident from equation (35), φ₅₂ _(_) ₂ is a combination (e.g., a sum) of an eighth magnetic flux φ_(3,3) flowing through the form 52 ₃ in a direction (e.g., bottom-to-top in FIG. 8) and a ninth magnetic flux −(φ_(1,3)+φ_(2,3)) simultaneously flowing through the form 52 ₃ in an opposite direction (e.g., top-to-bottom in FIG. 8).

From the foregoing it will be appreciated that, although specific embodiments have been described herein for purposes of illustration, various modifications may be made without deviating from the spirit and scope of the invention. Furthermore, where an alternative is disclosed for a particular embodiment, this alternative may also apply to other embodiments even if not specifically stated.

APPENDIX A

The following are mathematical expressions for parameters of a two-phase embodiment of the coupled-inductor core assembly 14 of FIGS. 1 and 4. These expressions have been derived using the magnetic circuit model 80 of FIG. 6.

L₁=Lc₁+L_(lk1) is the self inductance of the winding 16 ₁.

L₂=Lc₂+L_(lk2) is the self inductance of the winding 16 ₂.

L_(m)=Lc₁=Lc₂ is the coupling inductance between the windings 16 ₁ and 16 ₂.

L_(lk1) is the leakage inductance of the winding 16 ₁, and is defined in terms of the portion of the self flux L₁i₁ that flows through the leakage form 53 when i₁=1 A.

L_(lk2) is the leakage inductance of the winding 16 ₂, and is defined in terms of the portion of the self flux L₂i₂ that flows through the leakage form 53 when i₂=1 A.

R₁ is the reluctance of the form 52 ₁.

R₂ is the reluctance of the form 52 ₂.

R_(lk) is the reluctance of the leakage form 53.

A₁ is the cross-sectional area of the form 52 ₁.

A₂ is the cross-sectional area of the form 52 ₂.

A_(lk) is the cross-sectional area of the form 53.

μ₁ is the permeability of the form 52 ₁.

μ₂ is the permeability of the form 52 ₂.

μ₃ is the permeability of the form 53.

$L_{1} = \frac{N_{1}^{2}\left( {R_{2} + R_{k}} \right)}{{R_{1}R_{2}} + {R_{1}R_{k}} + {R_{2}R_{k}}}$ $L_{2} = \frac{N_{2}^{2}\left( {R_{1} + R_{k}} \right)}{{R_{1}R_{2}} + {R_{1}R_{k}} + {R_{2}R_{k}}}$ $L_{m} = \frac{N_{1}N_{2}R_{k}}{{R_{1}R_{2}} + {R_{1}R_{k}} + {R_{2}R_{k}}}$ $L_{k\; 1} = \frac{N_{1}^{2}R_{2}}{{R_{1}R_{2}} + {R_{1}R_{k}} + {R_{2}R_{k}}}$ $L_{k\; 2} = \frac{N_{2}^{2}R_{1}}{{R_{1}R_{2}} + {R_{1}R_{k}} + {R_{2}R_{k}}}$ $R_{1} = \frac{l_{1}}{\mu_{m\; 1} \cdot A_{1}}$ $R_{2} = \frac{l_{2}}{\mu_{m\; 2} \cdot A_{2}}$ $R_{k} = \frac{l_{k}}{\mu_{k}A_{k}}$

Special Examples

If core and winding If Leakage rung If core structure If winding structure structures are has no air gap is symmetrical is symmetrical both symmetrical (R_(k) = 0) (R₁ = R₂) (N₁ = N₂) (R₁ = R₂, N₁ = N₂) $L_{1} = \frac{N_{1}^{2}}{R_{1}}$ $L_{1} = \frac{2N_{1}^{2}}{R_{1} + {2R_{k}}}$ $L_{1} = \frac{N_{1}^{2}\left( {R_{2} + R_{k}} \right)}{{R_{1}R_{2}} + {R_{1}R_{k}} + {R_{2}R_{k}}}$ $L_{1} = \frac{2\; N_{1}^{2}}{R_{1} + {2R_{k}}}$ $L_{2\;} = \frac{N_{2}^{2}}{R_{2}}$ $L_{2} = \frac{2N_{2}^{2}}{R_{1} + {2R_{k}}}$ $L_{2} = \frac{N_{1}^{2}\left( {R_{1} + R_{k}} \right)}{{R_{1}R_{2}} + {R_{1}R_{k}} + {R_{2}R_{k}}}$ $L_{2} = \frac{2\; N_{1}^{2}}{R_{1} + {2R_{k}}}$ L_(m) = 0 $L_{m} = \frac{N_{1}N_{2}R_{k}}{R_{1}^{2} + {2R_{1}R_{k}}}$ $L_{m} = \frac{N_{1}^{2}R_{k}}{{R_{1}R_{2}} + {R_{1}R_{k}} + {R_{2}R_{k}}}$ $L_{m} = \frac{N_{1}^{2}R_{k}}{R_{1}^{2} + {2R_{1}R_{k}}}$ $L_{k\; 1} = \frac{N_{1}^{2}}{R_{1}}$ $L_{k\; 1} = \frac{N_{1}^{2}}{R_{1} + {2R_{k}}}$ $L_{k\; 1} = \frac{N_{1}^{2}R_{2}}{{R_{1}R_{2}} + {R_{1}R_{k}} + {R_{2}R_{k}}}$ $L_{k\; 1} = \frac{N_{1}^{2}}{R_{1} + {2R_{k}}}$ $L_{k\; 2} = \frac{N_{2}^{2}}{R_{2}}$ $L_{k\; 2} = \frac{N_{2}^{2}}{R_{1} + {2R_{k}}}$ $L_{k\; 2} = \frac{N_{1}^{2}R_{1}}{{R_{1}R_{2}} + {R_{1}R_{k}} + {R_{2}R_{k}}}$ $L_{k\; 2} = \frac{N_{1}^{2}}{R_{1} + {2R_{k}}}$ 

1.-21. (canceled)
 22. A method, comprising: driving a magnetizing first current through a first conductor and into an output node during a first period, the first conductor wrapped around a first form of a core, the magnetizing first current having a first magnitude and generating a first magnetic flux through the first form, the first form having a first characteristic of a first value, a density of the first magnetic flux flowing through the form depending on the characteristic; and directing a first portion of the first magnetic flux through a second form of the core, the first portion of the flux causing an induced second current to flow through a second conductor and into the output node, the second conductor wrapped around the second form of the core, the induced second current having a second magnitude, the second form having the characteristic of a second value that is different from the first value.
 23. The method of claim 22, further comprising directing a second portion of the magnetic flux through a third form of the core about which is wrapped no current-carrying conductor.
 24. The method of claim 22, further comprising directing a second portion of the magnetic flux through a third form of the core about which is wrapped no current-carrying conductor, the third form having the first characteristic of a third value.
 25. The method of claim 22, further comprising directing a second portion of the magnetic flux through a third form of the core about which is wrapped no current-carrying conductor, the third form having a second characteristic of a value that is different than values of the second characteristic for the first and second forms.
 26. The method of claim 22, further comprising: driving a magnetizing second current through the second conductor and into the output node during a second period, the magnetizing second current having a third magnitude and generating a second magnetic flux through the second form, a density of the second magnetic flux flowing through the second form depending on the characteristic; and directing a first portion of the second magnetic flux through the first form of the core, the first portion of the second flux causing an induced first current to flow through the first conductor and into the output node, the induced first current having a fourth magnitude.
 27. The method of claim 26 wherein: the third magnitude is substantially equal to the first magnitude; and the fourth magnitude is substantially equal to the second magnitude.
 28. A method, comprising: driving a magnetizing first current having a first magnitude through a first winding and into an output node during a first period, the first winding having a number of turns; and driving a magnetizing second current having a second magnitude through a second winding and into the output node during a second period, the second winding having the number of turns and being magnetically coupled to the first winding.
 29. The method of claim 28 wherein: the first winding is wound about a first form of a core, the magnetizing first current generating a first magnetic flux through the first form, the first form having a characteristic of a first value, a density of the magnetic flux flowing through the form depending on the characteristic; and the second winding is wound about a second form of the core, the magnetizing second current generating a second magnetic flux through the second form, the second form having the characteristic of a second value, a density of the magnetic flux flowing through the second form depending on the characteristic.
 30. The method of claim 22 wherein the first characteristic includes a length.
 31. The method of claim 22 wherein the first characteristic includes a cross-sectional area.
 32. The method of claim 22 wherein the first characteristic includes a magnetic permeability.
 33. The method of claim 29 wherein the characteristic includes a length.
 34. The method of claim 29 wherein the characteristic includes a cross-sectional area.
 35. The method of claim 29 wherein the characteristic includes a magnetic permeability. 